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Proof step 2 from proof induction (previous step)

μ(f)(x), monotone(f), ∀ x. f(λ x. μ(f)(x) ∧ φ(x))(x) → φ(x) ⊦ φ(x)

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 3:
⊦ μ(f)(x) ↔ (∀ S. f(S) ⊆ S → S(x))
Proof step 4:
μ(f)(x) ↔ (∀ S. f(S) ⊆ S → S(x)), μ(f)(x), monotone(f), ∀ x. f(λ x. μ(f)(x) ∧ φ(x))(x) → φ(x) ⊦ φ(x)

Rule argument

spec: monotone-set-fixpoint inst: name: least-prefixpoint-def seq: ⊦ μ(f)(x) ↔ (∀ S. f(S) ⊆ S → S(x)) subst: [f, x] → [f, x] flags: []

Additional information

Used terms: f, x